Violation of non-Abelian Bianchi identity and QCD topology

When Abelian monopoles due to violation of the non-Abelian Bianchi identity Jμ(x) condense in the vacuum, color confinement of QCD is realized by the Abelian dual Meissner effect. Moreover VNABI affects also topological features of QCD. Firstly, the topological charge density is not expressed by a total derivative of the Chern-Simons density Kμ(x), but has an additional term L(x)=2Tr(Jμ(x)Aμ(x)). Secondly, the axial U(1) anomaly is similarly modified, while keeping the Atiyah-Singer index theorem unchanged. However, if the integrated additional term $\mathrm{\Lambda }=(g^2\mathrm{/}16{\pi }^2)\int d^{4}xL(x)$ is not zero, it is not integer nor gauge invariant, so that VNABI would not be allowed in QCD. Using the Wu-Yang arguments, it is however proved that $\mathrm{\Lambda }$ becomes vanishing. $\mathrm{\Lambda }$ is evaluated also in the framework of Monte-Carlo simulations on SU(2) lattices in details with partial gauge fixings such as the Maximal Center gauge (MCG). When the gradient flow method is used, the term $\mathrm{\Lambda }$ tends to vanish after small gradient flow time ($\tau$). The biggest effect of VNABI on QCD topology seems to be that self-dual instantons can not be a classical solution of QCD at space-time points where VNABI occurs. One has to find an alternative mechanism explaining integer topological charge, etc. The bosonic definition of the topological charge $Q_t$ and its Abelian counterpart $Q_a\equiv (g^2\mathrm{/}16{\pi }^2)\int d^{4}x\text{Tr}(f_{\mu \nu }{f}_{\mu \nu }^{\ast })$ written by Abelian field strengths are measured also on the lattices. When $\chi$ is zero, $Q_a$=3$Q_t$ is expected theoretically.